Generalizations and Applications of the Lagrange Implicit Function Theorem
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The implicit function theorem due to Lagrange is generalized to enable high order implicit rate calculations of general implicit functions about pre-computed solutions of interest. The sensitivities thus calculated are subsequently used in determining neighboring solutions about an existing root (for algebraic systems) or trajectory (in case of dynamical systems). The generalization to dynamical systems, as a special case, enables the calculation of high order time varying sensitivities of the solutions of boundary value problems with respect to the parameters of the system model and/or functions describing the boundary condition. The generalizations thus realized are applied to various problems arising in trajectory optimization. It was found that useful information relating the neighboring extremal paths can be deduced from these implicit rates characterizing the behavior in the neighborhood of the existing solutions. The accuracy of solutions obtained is subsequently enhanced using an averaging scheme based on the Global Local Orthogonal Polynomial (GLO-MAP) weight functions developed by the first author to blend many local approximations in a continuous fashion. Example problems illustrate the wide applicability of the presented generalizations of Lagrange's classical results to static and dynamic optimization problems.